The fundamental solution to the paraxial wave equation (PWE), the Gaussian beam, encompasses important properties of the actual output of laser sources, such as their characteristic intensity profile and phase distribution, both of which evolve as the beam propagates.
One exact higher-order solution to the PWE is the Hermite-Gaussian (HG) mode, which is given in cartesian coordinates and is characterized by lobes of light disposed in a rectangular grid. HG modes are important in the study of laser cavities, as they describe the spatial distribution of the output beams from these light sources [1]}.
The Laguerre-Gaussian (LG) mode is another exact solution to the PWE. It is given in cylindrical coordinates and carries a ring-shaped intensity distribution.
Most importantly, LG modes carry well-defined orbital angular momentum (OAM) in the propagation direction per photon, as first demonstrated by Allen et al. [2]}.
HG and LG modes can be represented in terms of one another and this correspondence is well-known [2]}, [4]}. Also, they are limiting cases of the so-called Ince-Gaussian paraxial mode [5]}, that carries elliptical symmetry.
There are many other higher-order solutions to the PWE that present vastly diverse characteristics and interesting properties [6]}.
